Low-lying excited states of the eight-vertex model with an odd number of lattice sites

For the zero-field eight-vertex model in the disordered regime, the bulk free energy is known for both parities of the lattice size $N$: Baxter solved even $N$, and Pâtu (2007), using the 'generic' root equation discovered by Fabricius and McCoy for odd $N$, showed that the doubly degenerate largest eigenvalue yields the same thermodynamic free energy. The next tier of the spectral theory — the low-lying excitation spectrum for odd $N$, computed for even $N$ by Johnson, Krinsky and McCoy via linear integral equations — remains open. For crossing parameter $\eta = mK/L$ with $m$ odd the calculation is expected to be accessible: the roots of the auxiliary $Q$ matrix obey the generic equation and pair as $u$ and $u + iK'$, so the Johnson–Krinsky–McCoy integral-equation method must be adapted to this new root structure. For $m$ even and $L$ odd a deeper obstacle appears: some roots condense into $L$-strings that drop out of the generic equation while the remaining roots pair, the standard linear integral-equation method fails, the excitation spectrum contains two different kinds of states, and eigenvalues develop extra degeneracies beyond the universal double degeneracy of the odd case. The third open item of the source — the belief that at the special point $\eta = 2K/3$ the degenerate largest eigenvalue is exactly $(a+b)^N$ on finite odd lattices — was proven by Hagendorf and Liénardy in 2018 and is therefore excluded from the open core. The open problem is the odd-$N$ excitation program itself: derive and solve the appropriate integral (or replacement) equations, obtain the excitation gaps in the thermodynamic limit for both the $m$ odd and the $m$ even / $L$ odd cases, and characterize the two-class structure and its degeneracies.

Background

The eight-vertex model places arrows on the edges of the square lattice so that an even number of arrows points into each vertex (the 'ice rule'); the eight allowed vertex configurations receive Boltzmann weights $a, b, c, d$. In the zero-field case the weights are parametrized by a normalization $\rho$, a spectral parameter $v$, a crossing parameter $\eta$, and the elliptic modulus $k$ through products of Jacobi theta functions $H$ and $\Theta$; here $K = K(k)$ is the complete elliptic integral of the first kind and $K'$ its value at the complementary modulus. In the disordered regime the weights are positive. The row-to-row transfer matrix $T(v)$ of a lattice with $N$ columns and periodic boundary conditions has $2^N$ eigenvalues, and the bulk free energy per site is $f = -k_B T \lim_{N\to\infty} N^{-1}\ln|T_{\max}(N)|$.

Baxter solved the even-$N$ model in 1972 by introducing, at 'roots of unity' $\eta = mK/L$ (integers $m$ and $L$), an auxiliary matrix $Q_{72}(v)$ that commutes with $T(v)$ and satisfies the TQ equation $T(v)Q_{72}(v) = [\rho\Theta(0)h(v+\eta)]^N Q_{72}(v-2\eta) + [\rho\Theta(0)h(v-\eta)]^N Q_{72}(v+2\eta)$, where $h(v) = H(v)\Theta(v)$. Eigenvalues of $Q_{72}$ factorize as products of $H(v - u_k)$ over 'roots' $u_k$. Johnson, Krinsky and McCoy (1973) rederived Baxter's largest eigenvalue and computed the low-lying excitation spectrum using linear integral equations for the density of roots, and Klümper and Zittartz (1988, 1989) obtained all eigenvalues and classified the even-$N$ excited states. For even $N$ the roots appear either as 'Bethe pairs' $(u, u + iK')$ obeying Bethe's equation, or as $L$-strings $u + 2jK/L$, $j = 0,\ldots,L-1$, whose contribution cancels from the root equation.

For odd $N$ the structure is qualitatively different. Fabricius and McCoy (2005) found that the largest eigenvalue is doubly degenerate (the spin-inversion operator $S = \prod_j \sigma^z_j$ commutes with $T(v)$, and for odd $N$ the two spin-inversion sectors produce equal eigenvalues), and that when $m$ is odd — the case where $Q_{72}$ exists — the roots satisfy not Bethe's equation but a new 'generic equation', in which all $N$ roots appear and the right-hand side contains $H$ instead of $h$; every root $u$ is accompanied by a partner $u + iK'$. For $m$ even with $L$ odd, where no $Q_{72}$ matrix is known, numerical studies showed some roots condensing into $L$-strings that drop out of the generic equation while the remaining roots arrange in pairs. Pâtu (2007) used the generic equation to compute the bulk free energy for odd $N$, recovering the even-$N$ result in the thermodynamic limit, and posed the excitation problem stated below. At the special point $\eta = 2K/3$ (written $\eta = \pi/3$ in the normalization of Bazhanov and Mangazeev; the two conventions are related by $\eta_{\mathrm{BM}} = \pi\eta_B/(2K_B)$, where $\eta_B$ and $K_B$ denote Baxter's crossing parameter and complete elliptic integral), the degenerate largest eigenvalue was conjectured by Stroganov to be exactly $(a+b)^N$ on finite odd lattices; this is now a theorem (Hagendorf–Liénardy 2018) and no longer forms part of the open question.

Problem Statement

For the zero-field eight-vertex model in the disordered regime on a periodic lattice with an odd number of sites $N$, compute the low-lying excitation spectrum in the thermodynamic limit, generalizing the Johnson–Krinsky–McCoy treatment of even $N$. Concretely:

(i) For crossing parameter $\eta = mK/L$ with $m$ odd — where Baxter's $Q_{72}$ matrix exists and its roots obey the Fabricius–McCoy generic equation, each root $u$ being accompanied by a partner $u + iK'$ — derive and solve the linear integral equations for the root densities of the low-lying excited states, and obtain the excitation energies relative to the doubly degenerate largest eigenvalue, including their gaps in the $N \to \infty$ limit.

(ii) For $m$ even with $L$ odd — where some roots condense into $L$-strings that drop out of the generic equation while the remaining roots arrange in pairs, so that the standard linear integral equation method cannot be applied — develop the corresponding computation of the excitation energies, establish that the excitation spectrum contains two different kinds of states, and determine the degeneracy structure, including the extra degeneracies beyond the double degeneracy present in the odd case.

A complete answer supplies, for both the $m$ odd and the $m$ even / $L$ odd cases, the thermodynamic-limit excitation spectrum together with the degeneracy structure — including the two-class classification of excited states and its extra degeneracies in the $m$ even / $L$ odd case — and is consistent with the known even-$N$ spectrum and with the exactly known finite-lattice eigenvalue $(a+b)^N$ at $\eta = 2K/3$.

References

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